Finite Group Databasebeta

← #28362328

Subgroups — #28362328

Order 13952.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20295776C211yesmaximal, characteristic, complemented
2#20295390C212yesmaximal, complemented
3#20294767C212yesmaximal, complemented
4#20294766C211yesmaximal, characteristic, complemented
5#496×C109C211yesmaximal, characteristic
6#16812697—21no—
7#16812692—22no—
8#16812691C2212yes—
9#16812680—21no—
10#16812676—22no—
11#16812620—22no—
12#16812620—22no—
13#16812610C2212yes—
14C2×SD16×C109—21no—
15C2×D8×C109—21no—
16#127×C109C2211yescharacteristic
17#124×C109C2211yescharacteristic
18#97×C109C2211yescharacteristic
19C2×Dic218—21no—
20C22×Dic109—21no—
21C2×#11724784—22no—
22C4⋊Dic109D412yes—
23C4⋊Dic109D412yes—
24#13182008—22no—
25#13182008—22no—
26#13182008—22no—
27C22⋊Dic109—22no—
28C22⋊Dic109—21no—
29C4×Dic109—21no—
30C2×Q8×C109D411yescharacteristic
31C2×D4×C109—21no—
32C2×D4×C109—22no—
33C2×D4×C109D411yescharacteristic
34SD16×C109—42no—
35D8×C109—42no—
36C2×C8×C109—21no—
37C22⋊C4×C109—22no—
38C42×C109C2311yescharacteristic
39C23×C109—21no—
40C23×C109—41no—
41Q8×C109—41no—
42D4×C109—41no—
43D4×C109—41no—
44D4×C109—41no—
45D4×C109—44no—
46C2×C4×C109—21no—
47C2×C4×C109C2×D412yes—
48C2×C4×C109C2×D411yescharacteristic
49C2×Dic109—22no—
50C2×Dic109—22no—
51C2×Dic109—81no—
52Dic218—81no—
53C8×C109—22no—
54#11724784—42no—
55C22×C109—41no—
56C22×C109—42no—
57C22×C109—81no—
58C22×C109—82no—
59C22×C109#12011yescharacteristic
60C4×C109—21no—
61C4×C109—22no—
62C4×C109—41no—
63Dic109—42no—
64Dic109—42no—
65C2×C109—81no—
66C2×C109—161no—
67C2×C109#45311yescharacteristic
68C2×C109#45412yes—
69#1657—1091nomaximal, Sylow
70C109#165711yescharacteristic, complemented, Sylow
71#563—1091no—
72#496D10911yescharacteristic
73#490—1092no—
74#330—1092no—
75#329—1091no—
76C2×SD16—21no—
77C2×D8—21no—
78C4⋊Q8—2181no—
79#127C2×D10911yescharacteristic
80#126—2182no—
81#125—1092no—
82#124C2×D10911yescharacteristic
83#123—2181no—
84C22⋊Q8—2182no—
85#102—2182no—
86#102—2182no—
87#97—1092no—
88#97C2×D10911yescharacteristic
89C2×Q8—2181no—
90C2×Q8C22⋊D10911yescharacteristic
91C2×D4—22no—
92C2×D4—21no—
93C2×D4C22⋊D10911yescharacteristic
94C22×C4—2181no—
95SD16—42no—
96D8—42no—
97C2×C8—21no—
98C2×C8—2182no—
99C4⋊C4—1092no—
100C4⋊C4—1092no—
101C4⋊C4—2182no—
102C4⋊C4—2182no—
103C4⋊C4—2182no—
104C22⋊C4—22no—
105C22⋊C4—2182no—
106C22⋊C4—2181no—
107C42—2181no—
108C42C22×D10911yescharacteristic
109C23—21no—
110C23—41no—
111Q8—41no—
112Q8—8721no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—44no—
117C2×C4—21no—
118C2×C4—8721no—
119C2×C4—2182no—
120C2×C4—2182no—
121C2×C4D4×D10912yes—
122C2×C4C2×C22⋊D10911yescharacteristic
123C8—22no—
124C8—4362no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22#1681266711yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—4362no—
134C4—4362no—
135C2—81no—
136C2—161no—
137C2#2029519311yescharacteristic
138C2#2029519712yes—
subgroupG/H
centerC22#16812667
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting, prime radical#496×C109C2
socleC22×C109#120
hypercenter#496D109
hypocenterC109#1657
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20295776C2semidirect
2. #20295390C2semidirect
3. #20294767C2semidirect
4. #20294766C2semidirect
70. C109#1657semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202957762
#202953902
#202947672
#202947662
#496×C1092
#1657109